Structural stress analysis

ABSTRACT

Structural stress in a fatigue-prone region of a structure is determined and analyzed by using: i) the nodal forces and displacement values in the fatigue-prone region, or ii) equilibrium equivalent simple stress states consistent with elementary structural mechanics in the fatigue-prone region. Of course, it is contemplated that combinations, equivalents, or variations of the recited bases may alternatively be employed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.10/989,771 (BAT 0031 VA), filed Nov. 16, 2004, which is a division ofU.S. patent application Ser. No. 09/992,552 (BAT 0031 PA), filed Nov.16, 2001, which claims the benefit of U.S. Provisional Application Ser.No. 60/249,800 (BAT 0031 MA), filed Nov. 17, 2000.

BACKGROUND OF THE INVENTION

The present invention relates to structural stress analysis and, moreparticularly to stress calculations and a scheme for calculatingstructural stress where the structure contains geometricdiscontinuities, e.g., welded joints, notches, ridges, bends, sharpcorners, etc.

Stress analysis is used primarily in the design of solid structures suchas ships, automobiles, aircraft, buildings, bridges, and dams fordetermining structural behavior and for ascertaining structuralviability and integrity under anticipated or foreseeable loads. Theanalysis may involve the use of an abstract or mathematical model forthe representation of the structural system and loads. According toclassical analytical idealization, partial differential equations areused. For example, stress in a dam under a hydrostatic load can bedescribed by an elliptic partial differential equation in two spatialdimensions.

As boundary geometry of structural systems is usually complicated, thepartial differential equations of structural mechanics typically cannotbe solved in the closed analytical exact form. Numerical approximationsare sought instead. In one approach, derivatives are replaced withfinite differences. Other methods are based on finding an approximationas a linear combination of preassigned functions such as polynomials ortrigonometric functions. Also, after a domain or a boundary of interesthas been discretized in the form of a large number of small elements, apiece-wise approximation can be sought according to the finite elementmethod.

Current methods of stress analysis based upon numeric approximations andextrapolation are often subject to substantial uncertainties in regionsof close proximity to welds, joints, sudden changes of geometry, orother structural or geometric discontinuities and are highly dependenton element size and are typically mesh dependent, particularly ifdrastically different loading modes are considered. Accordingly, thereis a need for an improved structural stress analysis scheme for fatigueprediction that effectively eliminates or minimizes mesh dependency.

BRIEF SUMMARY OF THE INVENTION

This need is met by the present invention wherein structural stress in afatigue-prone region of a structure is determined and analyzed by using:i) the nodal forces and displacement values in the fatigue-prone region,or ii) equilibrium equivalent simple stress states consistent withelementary structural mechanics in the fatigue-prone region. Of course,it is contemplated that combinations, equivalents, or variations of therecited bases may alternatively be employed. The determination issubstantially independent of mesh size and is particularly well-suitedfor applications where S-N curves are used in weld fatigue design andevaluation, where S represents nominal stress or stress range and Nrepresents the number of cycles to failure.

Throughout the present specification and claims certain quantities,functions, parameters, and values are described or identified as beingdetermined, calculated, or measured. For the purposes of defining anddescribing the present invention, it is noted that the use of one ofthese terms herein is not intended to exclude steps that incorporate ormay be defined with reference to another of the terms. For example, a“determination” may incorporate aspects that are measured or calculated,a “calculation” may depend upon a specific measurement or determination,and a measurement may be based upon a calculation or determination. Theterm “analysis” is utilized herein to refer to an identification,review, or examination of the results of a determination, calculation,measurement, or combination thereof.

For the purposes of defining and describing the present invention, it isnoted that reference to a “mesh” or “grid” herein relates to the use ofany defined reference framework for identifying, quantifying, orotherwise defining individual portions of an object or an actual orvirtual model of an object. Typically, a single quantitative value isassociated with individual points along the reference framework or withindividual portions defined by the reference framework. A calculationthat is mesh insensitive is substantially unaffected by the inherentprecision, frequency, or definition of the reference framework. Instress/strain evaluation, it is common to utilize a finite element meshderived by dividing a physical domain into smaller sub-domains orelements in order to facilitate a numerical solution of a partialdifferential equation. Surface domains may be subdivided into triangleor quadrilateral shapes, while volumes may be subdivided primarily intotetrahedral or hexahedral shapes. The size, shape and distribution ofthe elements is ideally defined by automatic meshing algorithms.

The present invention is directed to a structural stress calculationscheme that generates stress determinations that are substantiallyindependent of the size, shape and distribution of the mesh elements.Validations of the stress analysis scheme of the present inventionreveal that structural stress determined according to the presentinvention proved to be substantially independent of mesh size over atleast the following mesh size ranges: (i) from about 0.16 t and 0.1 t,along respective x and y axes, to about 2 t and t along respective x andy axes; (ii) from about 0.5 t and t, along respective x and y axes, toabout 2 t and t along respective x and y axes; and (iii) from about0.008 t and 0.02 t, along respective x and y axes, to about 0.4 t and0.5 t along respective x and y axes, where t represents the thickness ofthe structure. For the purposes of defining the present invention, it isnoted that claims reciting a mesh are intended to cover any referenceframework, like a mesh, grid, or other framework, where a domain isdivided into sub-domains or elements.

The present invention has application to stress evaluation of a varietyof welded and non-welded joints, notches, ridges, bends, sharp corners,and other discontinuities or sudden changes in geometry in metallic,plastic and ceramic structures. Relevant structures include aircraft andaerospace equipment (aircraft, helicopters, rockets, gantries);agricultural equipment (aerators, balers, baggers, choppers, combines,cultivators, elevators, feeders, grain hoppers, bins and sorters,harrows, harvesters, mowers, mulchers, planters, plows, scrapers,seeders, shredders, sprayers, spreaders, tillers, threshers, tractors);agricultural structures (silos, barns, brooders, incubators);automobiles and trucks (automobiles, trucks, trailers, wagons, axles,hitches); construction and lifting equipment (bulldozers, cranes,hoists, winches, jacks, chains, spreaders, hi-los, backhoes, forklifts,loaders, haulers, scrapers, excavators, graders, trenchers, borers,directional drillers, pulverizers, saws); forestry equipment (skidders,feller bunchers, log loaders, log splitters); mining equipment; rail carframes; ships; submarines and submersibles; ports and port equipment(docks, piers, cranes, lighthouses); bridges (bridges, bridge expansionjoints); channels or canals; tunnels; building components (doors, doorframes, windows, sashes, shutters, soffits, fascia, skylights,scaffolding, cabinetry, chimneys and flues, elevators); buildingmaterials (framing channels and struts, joists, trusses and trussplates, wire reinforced concrete); appliances, home and industrial(sinks, stoves, ovens, refrigerators, baths); buildings/skyscrapers(steel decks, mezzanines); housing, particularly manufactured ormodular; heating and cooling systems (especially ducts); homeimprovement equipment (ladders, hand tools); fencing and gates; plumbing(pipes, fittings, pumps, sewage lines); irrigation and drainageequipment (pipes, irrigation sprinkler systems, drains and drainpipes);manufacturing equipment and machinery (conveyors, fasteners, riveters);diving equipment; nuclear containers and facilities; offshore oil rigs;diesel and gas turbines; pipelines; derricks and digger derricks;cooling towers; radio towers/transmitters; welded structures (weldedwire reinforcements); tanks and cisterns (esp. for water storage);aircraft components; automotive parts; footwear; household components(sinks, showers, plumbing pipe, swimming pools); sporting goods;ceramics; concrete; porcelain enamel; sealants and sealed structures,adhesively bonded structures; etc.

The stress analysis scheme of the present invention may also be utilizedto monitor stress, predict failure, schedule maintenance, determinefatigue, and analyze structures. The structural stress analysis schemeof the present invention is particularly significant in these contextsbecause a 10% variation in a stress value could translate into a lifecycle that varies by as much as 100-200%. Further, the stress analysisscheme of the present invention may be utilized as a structural designtool directed at optimizing manufacturing costs or as a tool formonitoring, locally or remotely, structural stress in situ or duringmanufacture or assembly to guide manufacturing or assembly steps.Structures may be analyzed by imaging the structure, creating a model ofthe structure and applying the principles of the present invention toassess the stress in the structure. It is contemplated that the schemeof the present invention may be incorporated into an otherwiseconventional, field-ready hand-held or desk top computer or programmabledevice.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The following detailed description of the preferred embodiments of thepresent invention can be best understood when read in conjunction withthe following drawings, where like structure is indicated with likereference numerals and in which:

FIGS. 1A, 1B, 1C, and 1D illustrate typical welded joints andcorresponding structural stress definitions and local stressdistributions associated therewith;

FIGS. 2A and 2B illustrate structural stress definitions and localstress distributions associated with spot and laser lap joints;

FIG. 3 illustrates normal and shear stress distributions from a typicalfinite element model at a reference cross section of a structure forstructural stress calculations;

FIG. 4 represents a structural stress definition, taken along sectionline A-A, corresponding to the normal and shear stress distributions ofFIG. 3;

FIG. 5A illustrates local normal stress and shear stress distributionsat a reference cross-section for a fatigue failure criterioncorresponding to a structure including partial thickness fatigue crack;

FIG. 5B represents the structural stress definition relative to theoverall thickness of the structure of FIG. 5A;

FIG. 5C represents the structural stress definition for the structure ofFIG. 5A with a partial thickness fatigue crack failure;

FIG. 6A illustrates local normal and shear stress distributions of astructure defining a non-monotonic through thickness stress distributionat a weld toe of the structure;

FIG. 6B illustrates respective local normal and shear stressdistributions at a reference section of the structure illustrated inFIG. 6A;

FIG. 6C illustrates structural stress components relative to both upperand lower portions of the structure illustrated in FIG. 6A;

FIG. 7 is a flow chart illustrating a method of calculating structuralstress for general three dimensional structures according to the presentinvention;

FIG. 8 illustrates a shell model and corresponding elements forstructural stress extraction according to one embodiment of the presentinvention;

FIG. 9 illustrates a solid model and corresponding elements forstructural stress extraction according to one embodiment of the presentinvention;

FIG. 10 illustrates a scheme for analyzing structural stress byutilizing stress resultants directly available from finite element codesor another source of similar data; and

FIGS. 11 and 12 illustrate experimental techniques for measuringstructural stress.

DETAILED DESCRIPTION

The following detailed description is presented according to thefollowing subject matter outline, as delineated in the body of thedescription:

-   1. Structural Stress Analysis by Enforcing Equilibrium Conditions    and solving for σ_(B), and-   1.1. Stress Analysis by Using Stress Distributions σ_(x)(y) and    τ_(xy)(y).-   1.2. Stress Analysis by Using Stress Resultants.-   1.3. Special Applications—Partial Thickness Fatigue Crack.-   1.4. Special Applications—Non-Monotonic Through-Thickness    Distributions.-   2.0. Calculation of Structural Stress by Conversion of Nodal Forces    and Moments.-   2.1. Conversion of Nodal Forces and Moments Retrieved Directly from    Shell Model.-   2.2. Conversion of Nodal Forces and Moments by Generating Stiffness    Matrices and Nodal Displacements from the Shell Model.-   2.3. Conversion of Nodal Forces and Moments from Three-Dimensional    Solid Model.-   3. Experimental Techniques for Measuring Structural Stress-   3.1. Monotonic Through-Thickness Distributions-   3.2. Non-monotonic Through-Thickness Stress Distributions

Referring initially to FIGS. 1A, 1B, 1C and 1D, it is noted thatstructures containing geometric discontinuities generally possesslocalized stress concentrations and corresponding through-thicknessstress distributions. For example, the structure or plate 10 illustratedin FIG. 1A includes a fillet weld 12 and defines a particularthrough-thickness stress distribution σ_(x)(y). The stress distributionσ_(x)(y) is defined along a cross section of the structure 10 in afatigue-prone region or weld toe 14 of the weld 12 and may be obtained,for example, from a finite element model of the structure 10. FIG. 1Billustrates the corresponding structural stress definition in terms oftwo components σ_(M), σ_(B) of the structural stress σ_(s) in thelocalized fatigue-prone region 24. The structural stress definitionillustrated in FIG. 1B is effectively a simple structural stressdistribution in the form of membrane and bending components σ_(M), σ_(B)that are equilibrium-equivalent to the local stress distributions inFIG. 1A.

Through-thickness stress distributions σ_(x)(y) with respect topotential failure planes for a structure 10′ including lap fillet joints12′ are illustrated in FIG. 1C. In such joint types, depending on actualgeometry and loading conditions, three possible fatigue-prone regions orfailure planes exist: (1) weld root failure along the weld throat; (2)weld toe failure; and (3) weld root/toe failure. The through-thicknessstress distributions σ_(x)(y) correspond to cross sections of thestructure 10′ taken along the different fatigue-prone regions of thestructure 10′. The corresponding equilibrium-equivalent simplestructural stress definitions are illustrated in FIG. 1D in terms of thetwo components σ_(M), σ_(B) of the structural stress σ_(s) in thedifferent fatigue-prone regions.

Referring now to FIGS. 2A and 2B, local normal stress distributionsσ_(x)(y) along a typical failure plane are illustrated for structures10″ including a spot weld 12″ (e.g., a resistance spot weld) or a laserlap weld 12″. Regardless of whether the structure 10″ includes a laserlap weld or resistance spot weld, the local normal stresses andstructural stresses are essentially defined the same. Note that forresistance spot welds, see FIG. 2A, both the local stresses and thestructural stresses are a function of circumferential angle. In the caseof typical laser lap welds, see FIG. 2B. In the case of a laser lapweld, the weld area will occupy the entire cross-section without thehalf-width (w/2). If the stress quantities can be adequately defined inthe x-y plane of FIG. 2B despite the fact that the long lap jointextends in a direction orthogonal to the x-y plane.

FIGS. 3 illustrates in further detail the local normal stressdistribution σ_(x)(y) and the shear stress distribution τ_(xy)(y) of astructure 20 or, more specifically, a structural member 20, in afatigue-prone region 24 in the vicinity of a weld 22 of the structure20. The values of σ_(x)(y) and τ_(xy)(y) are taken along line B-B ofFIG. 3 for calculation purposes only. The value of t corresponds to thethickness of the structure 20 in the selected cross section. Thestructural mid-plane 26 is illustrated in FIGS. 3 and 4. FIG. 4represents a structural stress definition corresponding to σ_(x)(y) andτ_(xy)(y) along section A-A of FIG. 3 and includes, more particularly,the two components σ_(M), σ_(B) of the structural stress σ_(s) in thelocalized fatigue-prone region 24. Also illustrated in FIGS. 3 and 4 isthe variable δ which represents the finite element size of a finiteelement model utilized in calculating structural stress according to thepresent invention. Typically, a single element row corresponds to thedistance between sections A-A and B-B. It is noted that σ_(B) assumes avalue of zero at the structural mid-plane 26.

1. Structural Stress Analysis by Enforcing Equilibrium Conditions andsolving for σ_(B), and σ_(M)

A finite element model of a structure can be used to retrieve eitherstress outputs or nodal forces and displacements of the structure. Ifnodal forces and displacements are retrieved from the model, they can beused to calculate structural stresses in an accurate manner. In usingtwo and three-dimensional solid element models, it can be moreconvenient to retrieve stress outputs directly from the finite elementmodel. Stress outputs retrieved in this manner must, however, beprocessed by enforcing equilibrium conditions at a reference location inthe structure.

According to the methodology of one embodiment of the present invention,structural stress σ_(s) in a localized fatigue-prone region of astructure (e.g., section A-A of FIGS. 3 and 4) is defined in terms ofthe bending and membrane components σ_(B), σ_(M) of the structuralstress σ_(s) in the region. The structural stress is calculated byenforcing equilibrium conditions in the region and performing a selectedintegration utilizing stress distributions σ_(x)(y) and τ_(xy)(y)derived from an established source, such as a finite element model.

As will be appreciated by those practicing the present invention, finiteelement models of a variety of structures have been, and may be,generated through computer-aided engineering. These finite elementmodels may be used to derive individual stresses or internal loads foran area of interest within the structure. Accordingly, the stressdistributions σ_(x)(y) and τ_(xy)(y) are typically derived from a finiteelement model and are described herein in the context of finite elementmodels. It is important to note, however, that the stress distributionsmay be derived or input from any of a variety of sources, eitherexisting or yet to be developed. For example, a number of finitedifference models and so-called meshless calculation systems arecurrently under development for structural stress modeling.

The structural stress distribution σ_(x)(y) must satisfy equilibriumconditions within the context of elementary structural mechanics theoryat both the hypothetical crack plane (e.g., at weld toe 14 in FIG. 1A)and a nearby reference plane, on which local stress distributions areknown a priori from typical finite element solutions. The uniqueness ofsuch a structural stress solution can be argued by considering the factthat the compatibility conditions of the corresponding finite elementsolutions are maintained at this location in such a calculation.

It should be noted that in typical finite element based stress analysis,the stress values within some distance from the weld toe can changesignificantly as the finite element mesh design changes. This behavioris referred to herein as mesh-size sensitivity. While local stressesnear a notch are mesh-size sensitive due to the asymptotic singularitybehavior as a notch position is approached, the imposition of theequilibrium conditions in the context of elementary structural mechanicsaccording to the present invention eliminates or minimizes the mesh-sizesensitivity in structural stress calculations. This is due to the factthat the local stress concentration close to a notch is dominated byself-equilibrating stress distribution.

1.1. Stress Analysis by Using Stress Distributions σ_(x)(y) andτ_(xy)(y)

According to one method of the present invention, a cross section ofinterest (e.g., section B-B of FIGS. 3 and 4), the location of which isdefined by a unit width, is selected such that the localized,fatigue-prone, region lies adjacent to the selected cross section. Forthe purposes of describing and defining the present invention, it isnoted that elements that are separated by a single element distance froma selected cross section are said to lie adjacent to the cross section.

A first component σ_(M) of the structural stress σ_(s) in the localizedregion (e.g., section A-A of FIGS. 3 and 4) represents what may bereferred to as a membrane component of the structural stress σ_(s). Asecond component σ_(B) of the structural stress σ_(s) in the localizedregion represents what may be referred to as a bending component of thestructural stress σ_(s). The structural stress σ_(s) is calculated bycombining the first component σ_(M) of the structural stress and thesecond component σ_(B) of the structural stress as follows:σ_(s)=σ_(M)+σ_(B)

By imposing equilibrium conditions between Sections A-A and B-B, thestructural stress components σ_(m) and σ_(b) may be determined accordingto the following equations. Specifically, the first component σ_(M) ofthe structural stress as in the localized region is determined accordingto the following equation:$\sigma_{M} = {\frac{1}{t}{\int_{0}^{t}{{\sigma_{x}(y)}\quad{\mathbb{d}y}}}}$

The stress distribution σ_(x)(y) along the selected cross sectionrepresents the local structural through-thickness stress distributionand is determined from the finite element model of the structure. Thesecond component σ_(B) of the structural stress as in the localizedregion is determined by solving the following equation for σ_(B):${{\left( \frac{t^{2}}{2} \right)\sigma_{M}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {{\int_{0}^{t}{{\sigma_{x}(y)}y\quad{\mathbb{d}y}}} + {\delta{\int_{0}^{t}{{\tau_{xy}(y)}\quad{\mathbb{d}y}}}}}$

The term y corresponds to a distance along the y-axis from y=0 to amaterial point of interest in the selected cross section. The term tcorresponds to the thickness of the structure in the selected crosssection and δ represents the element size of the finite element model.The function τ_(xy)(y) represents the shear stress distribution and isdetermined from the finite element model of the structure. It iscontemplated that, in practice, a row of elements may be used betweenthe localized fatigue-prone region of a structure (e.g., section A-A ofFIGS. 3 and 4) and the selected cross section (e.g., section B-B ofFIGS. 3 and 4).

The equation for the first component σ_(M) of the structural stressσ_(s) represents the force balances in the x direction, evaluated alongB-B. The equation for the second component σ_(B) of the structuralstress σ_(s) represents the moment balances with respect to Section A-Aat y=0. The integral term on the right hand side of the equation for thesecond component σ_(B) represents the transverse shear force as acomponent of the structural stress definition. Accordingly, it is notedthat, under certain conditions, such as if the transverse shear effectsare not significant or if the finite element formulation providesadequate consideration of shear locking effects (particularly when usingrelatively large size elements), the structural stresses can becalculated directly from section A-A as follows:$\sigma_{M} = {{{\frac{1}{t}{\int_{0}^{t}{{\sigma_{X}(y)}\quad{\mathbb{d}{y\left( \frac{t^{2}}{2} \right)}}\sigma_{M}}}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {\int_{0}^{t}{{\sigma_{X}(y)}y\quad{\mathbb{d}y}}}}$

1.2. Stress Analysis by Using Stress Resultants.

Referring briefly to FIG. 10, stresses and nodal quantities from shellor plate finite element models are often defined in a global coordinatesystem (x, y, z), depending on the finite element codes used. Given thedefinition of the structural stress components in Eq. 1, it is the localcoordinate system (x′,y′,z′) that is convenient for calculating thestructural stresses with respect to a weld, with local x′ and y′ beingperpendicular and parallel to the weld direction, respectively.Consistent with the solid element model approach (e.g., see FIG. 3),three components of the stress resultants (sectional forces andmoments), i.e., f_(x′), f_(z′), and m_(y′), at Section B-B in FIG. 10can be used to calculate the structural stress components at SectionA-A: it is noted that structural stress may also be analyzed byutilizing stress resultants directly available from finite element codesor another source of similar data. Specifically, three components of thestress resultants f_(x′), f_(z′), and m_(y′) in at Section B-B in FIG.10 can be used to calculate the structural stress components at SectionA-A according to the following equation:$\sigma_{s} = {{\sigma_{M} + \sigma_{B}} = {\frac{f_{x^{\prime}}}{t} + \frac{6\left( {m_{y^{\prime}} + {\delta\quad f_{z^{\prime}}}} \right)}{t^{2}}}}$where δ and t represent the dimensional values illustrated in FIG. 10and f_(x′), f_(z′), and m_(y′) represent the stress resultantsillustrated in FIG. 10.

According to this aspect of the present invention, a finite elementformulation with six degrees of freedom at each node is assumed, i.e.,six components of generalized forces at each node (three translationaland three rotational). If stresses in the global coordinate system areused, they must be transformed to the local coordinate system beforestructural stress is calculated in the manner described above.

1.3. Special Applications—Partial Thickness Fatigue Crack.

Referring to FIGS. 5A, 5B, and 5C, an alternative scheme is describedherein for calculating structural stress σ_(x) where the selected crosssection of the structure at issue includes a partial thickness fatiguecrack extending a distance t₁ below a surface of the structure.According to this alternative scheme, the stress calculation schemediscussed above with respect to FIGS. 3 and 4, is altered by introducingtwo additional structural stress parameters σ_(m)′ and σ_(b)′ to enablethe calculation of structural stress σ_(s) where the selected crosssection of the structure at issue includes the partial thickness fatiguecrack extending a distance t₁ below a surface of the structure 20. Theparameters σ_(m)′ and σ_(b)′ are illustrated in FIG. 5C. The overallthrough-thickness structural stress components σ_(m) and σ_(b),illustrated in FIG. 5B, are calculated as discussed above with respectto the components σ_(M) and σ_(B) of FIGS. 3 and 4. For the scheme ofFIGS. 5A, 5B, and 5C, the structural stress σ_(s) is calculated bycombining a redefined first component σ_(M) of the structural stress anda redefined second component σ_(B) of the structural stress as follows:σ_(s)=σ_(M)+σ_(B)According to this alternative scheme, the first and second componentsσ_(M), σ_(B) of the structural stress σ_(s) in the localized region,with respect to t₁ are determined by solving a first equation throughdirect integration and by solving simultaneously three equations withthree unknowns.

A first equation of the four equations defines the sub-component σ_(m)′of the structural stress σ_(s). A second equation of the four equationsdefines a force equilibrium condition. A third equation of the fourequations defines a moment equilibrium condition. A fourth equation ofthe four equations defines a stress continuity condition.

Regarding the first equation, since σ_(m)′ is not mesh-size sensitive,it can be calculated by direct integration along B-B within the lengthof t−t₁ as follows:$\sigma_{M}^{\prime} = {{\frac{1}{t - t_{1}}{\int_{0}^{t - t_{1}}{{\sigma_{x}(y)}\quad{\mathbb{d}y}}}} - {\frac{1}{t - t_{1}}{\int_{0}^{\delta}{{\tau_{yx}(x)}\quad{\mathbb{d}x}}}}}$where σ_(m)′ and σ_(b)′ comprise respective sub-components of thestructural stress σ_(s) and are taken relative to a reference line 26′of the structure 20. The position of the reference line 26′ correspondsto the centroid or neutral axis of the region defined between t=0 andt=t−t₁. The structural mid-plane 26 is also illustrated.

By enforcing force balances along x and y, and moment balance at y=0 atSection A-A, the second, third, and fourth equations are available. Thesecond equation, defining the force equilibrium condition along thex-axis, is as follows:σ_(M) t ₁+σ_(m)′(t−t ₁)=σ_(m) t.

The third equation, defining the moment equilibrium condition is asfollows:${{\left( {\sigma_{M} - \sigma_{B}} \right){t_{1}\left( {t - \frac{t_{1}}{2}} \right)}} + {\sigma_{B}{t_{1}\left( {t - \frac{t_{1}}{3}} \right)}} + {\sigma_{m}^{\prime}\frac{\left( {t - t_{1}} \right)^{2}}{2}} + {\sigma_{b}^{\prime}\frac{\left( {t - t_{1}} \right)^{2}}{6}}} = {{\sigma_{m}\left( \frac{t_{1}^{2}}{2} \right)} + {{\sigma_{b}\left( \frac{t_{1}^{2}}{6} \right)}.}}$

The fourth equation, defining the stress continuity condition at y=t−t₁is as follows:σ_(M)−σ_(B)=σ_(m)′+σ_(b)′

These three additional equations, defining three unknowns (σ_(M), σ_(B),σ_(b)′), may be solved simultaneously, according to conventionalmethods, to determine the values of the first and second componentsσ_(M), σ_(B) of the structural stress σ_(s) in the localized region.

1.4. Special Applications—Non-Monotonic Through-Thickness Distributions.

Referring now to FIGS. 6A, 6B, and 6C, a process for calculatingstructural stress in a localized fatigue-prone region of a structure 30defining a non-monotonic through thickness stress distribution isillustrated. The structure 30 includes a pair of welds 32 defining weldtoes 34 and is merely illustrated partially and schematically in FIGS.6A, 6B, and 6C. FIG. 6A illustrates local normal and shear stressdistributions of the structure 30 at the weld toe 34 of the structure30. FIG. 6B illustrates respective local normal σ_(x)(y) and shearτ_(xy)(y) stress distributions at a reference section of the structure30. FIG. 6C illustrates structural stress components relative to bothupper and lower portions of the structure 30.

The stress distribution σ_(x)(y) of FIGS. 6A and 6B defines a minimumaxial stress along a secondary axis 36′ of the structure 30 where thetransverse shear stress distribution τ_(xy)(y) changes signs. Thesecondary axis 36′ of the structure lies a distance t₂ below a surfaceof the structure 30. The structural mid-plane 36 is also illustrated inFIGS. 6A, 6B, and 6C. The cross-section references A-A and B-B aresimilar to those described above with reference to FIGS. 3 and 4. Theparameter t₂ can be determined based on the position at which thetransverse shear stress changes direction, if there is no specifiedcrack depth as a failure criterion.

Two additional structural stress parameters σ_(m)′ and σ_(b)′ areillustrated in FIG. 6C. FIG. 6C illustrates these further sub-componentsσ_(m)′ and σ_(b)′ of structural stress σ_(s) taken relative to thesecondary axis 36′ of the structure 30. FIG. 6C also illustrates the twoprimary components σ_(M) and σ_(B) of the structural stress σ_(s) whereσ_(S)=σ_(M)+ν_(B)and where the stress continuity condition at y=t−t₂ is as follows:σ_(M)−σ_(B)=σ_(m)′+σ_(b)′

FIG. 6C also illustrates the structural stress definitions correspondingto the non-monotonic through-thickness stress distribution illustratedin FIG. 6A. If the stress distribution σ_(x)(y) is symmetric withrespect to the mid-plane 36 of the structural member 30, then t₂=t/2;and the reference plane 36′ coincides with the mid-plane 36. Generally,σ_(M), σ_(B), σ_(b)′, and σ_(m)′ can be solved in a manner similar tothe scheme illustrated above with respect to the embodiment of FIGS. 5A,5B, and 5C, provided t₂ has been determined. The parameter t₂ can bedetermined either by considering equilibrium conditions with respect tothe entire through-thickness section (t) or, more conveniently, byidentifying the point at which the transverse shear stress distributionτ_(xy)(y) changes signs, as is also discussed above with reference tothe location of the structural mid-plane 26 of FIG. 5B.

Where the top surface of the structure 30 corresponds to the location ofthe maximum structural stress, the structural stress σ_(s) in thelocalized region may be calculated by solving simultaneously thefollowing equations for the two unknowns, σ_(M), σ_(B):$\sigma_{M} = {{{\frac{1}{t_{2}}{\int_{t - t_{2}}^{t}{{\sigma_{X}\quad(y)}{\mathbb{d}{y\left( \frac{t_{2}^{2}}{2} \right)}}\sigma_{M}}}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {{\int_{t - t_{2}}^{t}{{\sigma_{X}(y)}y\quad{\mathbb{d}y}}} + {\delta{\int_{t - t_{2}}^{t}{{\tau_{xy}(y)}\quad{\mathbb{d}y}}}}}}$

The two additional structural stress parameters σ_(m)′ and σ_(b)′ can becalculated according to the following equations:$\sigma_{m}^{\prime} = {\frac{1}{t - t_{2}}{\int_{0}^{t - t_{2}}{{\sigma_{X}(y)}\quad{\mathbb{d}y}}}}$${{\sigma_{m}^{\prime}\frac{\left( {t - t_{2}} \right)^{2}}{2}} + {\sigma_{b}^{\prime}\frac{\left( {t - t_{2}} \right)^{2}}{6}}} = {{\int_{0}^{t - t_{2}}{{\sigma_{X}(y)}y\quad{\mathbb{d}y}}} + {\delta{\int_{0}^{t - t_{2}}{{\tau_{xy}(y)}\quad{\mathbb{d}y}}}}}$

For the calculation of structural stress for resistance spot and laserlap welds (see FIGS. 2A and 2B), if two or three-dimensional solidelement models are used, the calculation procedures are similar to theprocedures described above with reference to FIG. 4. The fatigue cracktypically initiates at the weld edge at the interface between the twosheets, where structural stress peak is located. A special case ariseswhere the joint configuration and loading are symmetric with respect theneutral axis. In this case, the shear stress on the cross section alongthe symmetry line is zero and structural stress components may becalculated by substituting t₂=t/2 and τ_(yx)=0.

2.0. Calculation of Structural Stress by Conversion of Nodal Forces andMoments.

According to another embodiment of the present invention, structuralstress σ_(s) in a localized fatigue-prone region of a structure is alsocalculated. This embodiment of the present invention involves conversionof relevant nodal forces and moments, or internal force and moments atthe nodes of a group of elements aligned with a weld toe line, tosectional forces and moments and may or may not involve a directintegration step similar to that described herein with reference toFIGS. 3-6.

As will be appreciated by those practicing the present invention, finiteelement models of a variety of structures have been, and may be,generated through computer-aided engineering. These finite elementmodels may be used to derive individual stresses for an area of interestwithin the structure. Accordingly, the nodal forces and moments aretypically derived from a finite element model and are described hereinin the context of finite element models. It is important to note,however, that the nodal forces and moments may be derived or input fromany of a variety of sources, either existing or yet to be developed. Forexample, a number of finite difference models and so-called meshlesscalculation systems are currently under development for structuralmodeling.

2.1. Conversion of Nodal Forces and Moments Retrieved Directly fromShell Model.

Specifically, referring to FIGS. 7 and 8, according to one aspect of thepresent invention, structural stress σ_(s) in a localized fatigue-proneregion of a structure is calculated from a finite element shell model ofthe structure. Typically, as is illustrated in FIG. 8, the structure andthe welds are modeled with four-node (quadrilateral) shell or plateelements 42. Initially, a shell element model 40, see FIG. 8, of thestructure at issue is generated utilizing a conventional computer aidedengineering application (see step 100). Finite element analysis isperformed on the shell element model (see step 102) and a determinationis made as to whether nodal forces and moments for local elements ofinterest 42 can be directly retrieved from the output of the finiteelement analysis (see step 104). As is illustrated in FIG. 8, theelements 42 are positioned adjacent to a localized fatigue-prone regionof a weld 44 and, as such, are identified for structural stressextraction. If the nodal force and moment vectors for the local elements42 may be retrieved directly from the finite element shell model 40 thenselected ones of the nodal force and moment vectors are converted tosectional force vectors n and moment vectors m with an appropriatemapping function that provides the consistent nodal loads, or workequivalent (see step 112). The conversion is performed in a workequivalent manner with respect to said nodal displacements determinedfor said nodal force and moment vectors (see step 113). Specifically,the conversion is performed such that a quantity of work correspondingto said nodal displacements and nodal force and moment vectors isequivalent to a quantity of work for said nodal displacements and saidsectional force and moment vectors n and m.

The mapping function ideally should be the same as that used in thefinite element program used in performing the structural analysis. Themapping function provides sectional forces (force per unit length) andmoments (moment per unit length) corresponding to the nodal forces andmoments. If high stress gradients exist, after mapping, the sectionalforces and moments for a group of elements along the weld may not becontinuous from one element to another. However, the total nodal forcesand moments (as well as displacements) shared by neighboring elements ata nodal point or a grid point are correct from typical finite elementsolutions. The continuity in sectional forces and moments must beenforced by considering the neighboring elements and solving a system oflinear equations formulated by using an appropriate mapping function foreach element. The unknowns in the system of linear equations aresectional forces and moments at the nodes adjoining neighboringelements. The system of linear equations are formulated using nodalforces and moments (outputs from a finite element shell or plate models)and element geometric information. After solving the system of linearequations, the sectional forces and moments in the direction of interestcan be used to calculate the structural stresses to be described below.If stress gradients along weld lines are not high and element shapes arenot distorted, the structural stresses can be directly calculated on anelement-by-element basis without resort to solving a system of linearequations, as discussed below with reference to the equation forstructural stress σ_(s).

Structural stress σ_(s) in a localized fatigue-prone region of thestructure may then be calculated according to the equationσ_(s)=σ_(B)+σ_(M)where σ_(B) is proportional to the sectional moment vector m and am isproportional to the sectional force vector n. More specifically, thestructural stress as may be calculated utilizing the following equation:$\sigma_{s} = {{\sigma_{B} + \sigma_{M}} = {\frac{12{mz}}{t^{3}} + \frac{n}{t}}}$where t corresponds to the thickness of the structure in thefatigue-prone region and z ranges from +t/2 at the top surface of thestructure to −t/2 at the bottom surface of the structure.

2.2. Conversion of Nodal Forces and Moments Using Stiffness Matrices andNodal Displacements from the Shell Model.

In some applications, the reference section B-B in FIG. 10 may not beeasily defined. This situation arises if welds are rather close to eachother or load transfer at a weld of interest is very localized. If theelement sectional forces and moments (with respect to the referenceelement in FIG. 10) at Section A-A are available from a finite elementsolution, the equilibrium requirements are automatically satisfiedwithin the accuracy of the finite element solutions. In this context, ifthe nodal force and moment vectors can not be retrieved directly fromthe finite element model of the structure (see step 104) then the nodalforce and moment vectors are computed by generating stiffness matricesand nodal displacements for the local elements in the fatigue-proneregion from the finite element model (see step 106). The nodal force andmoment vectors for elements of interest may then be computed bymultiplying the stiffness matrices and nodal displacements to obtainglobal nodal force and moment vectors at nodal points of said localelements and transforming the resulting global force and moment vectorsfrom the global coordinate system to the local coordinate system of anelement of interest (see step 108). As is noted above, thetransformation is performed in a work equivalent manner with respect tosaid nodal displacements (see step 109).

More specifically, the global stiffness matrix and nodal displacementsmay be used to determine local nodal force and moment vectors of anelement of interest because the global element stiffness matrices andnodal displacements of the structure at issue are known (see step 110).The structural stress is calculated as follows:$\sigma_{s} = {{\sigma_{M} + \sigma_{B}} = {\frac{f_{x^{\prime}}}{t} + \frac{6\left( m_{y^{\prime}} \right)}{t^{2}}}}$where t represents the dimensional value illustrated in FIG. 10 andf_(x′) and m_(y′) represent the stress resultants illustrated in FIG.10.

It should be noted that, according to this aspect of the presentinvention, the finite element software used must allow for generation ofthe stiffness matrices and the nodal displacements for the elements ofinterest. It should also be noted that, since the element nodal forcesand moments are based on finite element solutions, the transverse sheareffects considered earlier in the context of two-dimensional solidmodels are already taken into account in the shell element approach.

2.3. Conversion of Nodal Forces and Moments from Three-Dimensional SolidModel.

As is illustrated in FIG. 9, structural stress σ_(s) in a localizedfatigue-prone region of a structure may also be calculated from a finiteelement solid model of the structure by using nodal forces in a mannersimilar to that described above in the context of the shell/plateelement models (see FIGS. 7 and 8). A general three dimensional finiteelement solid model 40′ including local elements 42′ and a weld 44 isillustrated in FIG. 9. A fatigue-prone plane of interest is illustratedalong section A-B-C. Nodal forces on element faces can be eitherdetermined directly or through the local element stiffness matrix in amanner similar to that described in the context of FIG. 7. Once thenodal forces at the nodal positions are obtained, equivalent sectionalforces and moments along Section A-B-C can then be obtained usingconsistent mapping or shape functions. The structural stress can thencalculated with the equivalent sectional forces and moments in themanner described with reference to FIGS. 7 and 8. In addition,equivalent transverse shear effects need to be considered as discussedearlier with respect to the two-dimensional solid models.

It is contemplated that the various calculation schemes of the presentinvention are best performed with the aid of a computer programmed toexecute the variety of calculation steps described herein or a computerarranged to read a suitable program stored on an appropriate recordingmedium.

3. Experimental Techniques for Measuring Structural Stress

Referring now to FIGS. 11 and 12, it is noted that the structural stresscalculation schemes of the present invention may also be utilized infashioning experimental techniques for measuring structural stress byenforcing equilibrium conditions within a strip element adjacent to theweld toe of interest.

3.1. Monotonic Through-Thickness Distributions

FIG. 11 illustrates the local normal stress distribution σ_(x)(y) of astructure 20 or, more specifically, a structural member 20, in afatigue-prone region 24 in the vicinity of a weld 22 of the structure20. The value of σ_(x)(y) is taken along line A-A. The value of tcorresponds to the thickness of the structure 20 in the selected crosssection. The structural mid-plane 26 is illustrated in FIG. 11. Straingauges 60, or other displacement measuring devices, are arranged alongcross-sectional lines B-B and C-C.

Where through-thickness normal stress distributions take the form asshown in FIG. 11, stresses at the strain gauges can be calculated usingstrain gauge readings from measured positions along sections B-B andC-C. If the through-thickness stress distributions at the measuredpositions are approximately linear then the stress measurements can bedecomposed as follows:$\sigma_{b}^{B} = {\frac{1}{2}\left( {\sigma_{T}^{B} - \sigma_{B}^{B}} \right)}$$\sigma_{b}^{C} = {\frac{1}{2}\left( {\sigma_{T}^{C} - \sigma_{B}^{C}} \right)}$where the superscripts B and C indicate the sections at which thestresses are measured and subscripts T and B refer the values at the topand bottom surfaces, respectively. Linearity of the stress distributionsat the measure positions can be readily confirmed by using one or morestrain gauge readings, either at the top or bottom, to see if a lineardistribution along the surface is developed.

The transverse shear resultant force effects at B-B, as describedearlier in the context of finite element models, can be approximated by$F_{t} = {\frac{1}{l}\frac{I}{\left( {t/2} \right)}\left( {\sigma_{b}^{C} - \sigma_{b}^{B}} \right)}$where I is the moment of inertia of the structural member 20. If thedistance between sections A-A and C-C is small, the increase in thebending stress component at section A-A can be approximated as:$\sigma_{b} = {\sigma_{b}^{B} + {\frac{L}{l}\left( {\sigma_{b}^{C} - \sigma_{b}^{B}} \right)}}$The structural stress, as defined previously in FIG. 4, can then beapproximated based on the measurements at sections B-B and C-C as:$\sigma_{S} = {\sigma_{T}^{B} + {\frac{L}{l}\left( {\sigma_{b}^{C} - \sigma_{b}^{B}} \right)}}$The distances among sections A-A, B-B, and C-C are typically measured interms of fractions of thickness t.

3.2. Non-Monotonic Through-Thickness Stress Distributions

Referring now to FIG. 12, it is noted that non-monotonic stressdistributions can be handled in a manner similar to that described abovewith reference to FIG. 11. FIG. 12 illustrates the local normal stressdistribution σ_(x)(y) of a structure 30 or, more specifically, astructural member 30, in a fatigue-prone region 44 in the vicinity of aweld 32 of the structure 30. The value of σ_(x)(y) is taken along lineA-A. The value of t corresponds to the thickness of the structure 30 inthe selected cross section. The structural mid-plane 36 is illustratedin FIG. 11. Strain gauges 60 are arranged along cross-sectional linesB-B, C-C, and D-D.

Measurements at section D-D, where the stress distribution σ_(x)(y)′becomes approximately linear, provide the through-thickness mean stressσ_(m) as follows:$\sigma_{m} = \frac{\left( {\sigma_{T}^{D} + \sigma_{B}^{D}} \right)}{2}$Typically, σ_(m)=F/A (load F and cross-sectional area A are typicallyavailable). Accordingly, the structural stress component σ_(M), asdefined in FIG. 6C, becomes σ_(M)=σ_(m) andσ_(b) ^(B)=σ_(T) ^(B)−σ_(m)σ_(b) ^(C)=σ_(T) ^(C)−σ_(m)The structural stresses with respect to the upper half of the thickness(as shown in FIG. 6C) can be approximated as:$\sigma_{S} = {\sigma_{T}^{B} + {\frac{L}{l}\left( {\sigma_{b}^{C} - \sigma_{b}^{B}} \right)}}$

It is noted that general monotonic and symmetric stress distributionscan be handled in a manner to that illustrated in FIGS. 11 and 12, withthe exception that t₂ and the corresponding appropriate equilibriumconditions must be accounted for in the manner discussed above withrespect to FIG. 6C. A structure including a partial thickness fatiguecrack, as illustrated with reference to FIGS. 5A-5C above, can also betreated with a technique similar to that illustrated with reference toFIG. 11, as long as the two strain gauges 60 utilized at the bottom ofthe structure 20 (see FIG. 11) are positioned a distance t₁ from the topedge (see FIG. 5B). The specific calculation procedures are then thesame as those illustrated with reference to the monotonicthrough-thickness distribution case of FIG. 11.

It is also noted that the present application discusses conventionalstrain gauges as a suitable means for measuring displacement/strainmerely for the purposes of illustrating the present invention. It iscontemplated that a variety of displacement/strain measurement devicesmay be employed according to the present invention including, but notlimited to, imaging devices, laser-based devices, fiber-optic gauges,ultrasonic gauges, etc.

For the purposes of describing and defining the present invention it isnoted that the term “substantially” is utilized herein to represent theinherent degree of uncertainty that may be attributed to anyquantitative comparison, value, measurement, or other representation.The term “substantially” is also utilized herein to represent the degreeby which a quantitative representation may vary from a stated referencewithout resulting in a change in the basic function of the subjectmatter at issue.

As will be appreciated by those practicing the present invention, it ispossible to utilize alternatives to the selected integrations describedin detail herein. Indeed, any mathematical or procedural operation thathas a result that is substantially equivalent to the result of aselected integration may be performed according to the presentinvention. For the purpose of defining the present invention, a claimrecitation of an operation having a result substantially equivalent to aresult of a given integration reads on the integration itself and theequivalent operation.

Having described the invention in detail and by reference to preferredembodiments thereof, it will be apparent that modifications andvariations are possible without departing from the scope of theinvention defined in the appended claims. More specifically, althoughsome aspects of the present invention are identified herein as preferredor particularly advantageous, it is contemplated that the presentinvention is not necessarily limited to these preferred aspects of theinvention.

1. A method of analyzing structural stress σ_(s) in a fatigue-proneregion of a structure, said method comprising: determining a stressdistribution σ_(x)(y) along a selected cross section of said structure;determining a first component σ_(M) of said structural stress σ_(s) insaid fatigue-prone region by performing an operation having a resultsubstantially equivalent to a result of the following first integration$\sigma_{M} = {\frac{1}{t}{\int_{0}^{t}{{\sigma_{X}(y)}\quad{\mathbb{d}y}}}}$where σ_(x)(y) represents said through-thickness stress distribution andt corresponds to a thickness of said structure; determining a secondcomponent σ_(B) of said structural stress σ_(s) in said fatigue-proneregion by performing an operation having a result substantiallyequivalent to a solution of the following equation for σ_(B)${{\left( \frac{t^{2}}{2} \right)\sigma_{M}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {\int_{0}^{t}{{\sigma_{X}(y)}y\quad{\mathbb{d}y}}}$or, its mathematical equivalent$\sigma_{B} = {\frac{6}{t^{2}}{\int_{0}^{t}{{\sigma_{X}(y)}\left( {y - \frac{t}{2}} \right)\quad{\mathbb{d}y}}}}$where y corresponds to a position along said selected cross section, tcorresponds to a thickness of said structure, and σ_(x)(y) representssaid through-thickness stress distribution; and calculating saidstructural stress σ_(s) by combining said first component σ_(M) of saidstructural stress and said second component σ_(B) of said structuralstress.
 2. A method of analyzing structural stress as claimed in claim 1wherein said second component σ_(B) of said structural stress σ_(s) insaid fatigue-prone region is determined by solving the followingequation for σ_(B)${{\left( \frac{t^{2}}{2} \right)\sigma_{M}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {\int_{0}^{t}{{\sigma_{X}(y)}y\quad{{\mathbb{d}y}.}}}$3. A method of analyzing structural stress as claimed in claim 1 whereinsaid second component σ_(B) of said structural stress σ_(s) in saidfatigue-prone region is determined by solving the following equation forσ_(B)$\sigma_{B} = {\frac{6}{t^{2}}{\int_{0}^{t}{{\sigma_{X}(y)}\left( {y - \frac{t}{2}} \right)\quad{\mathbb{d}y}}}}$4. A computer-readable medium encoded with a computer program foranalyzing structural stress σ_(s) in a fatigue-prone region of astructure according to the method of claim
 1. 5. A system for analyzingstructural stress σ_(s) in a fatigue-prone region of a structureaccording to the method of claim
 1. 6. A method of analyzing structuralstress σ_(s) in a fatigue-prone region of a structure, said methodcomprising: determining a stress distribution σ_(x)(y) along a selectedcross section of said structure; determining a first component σ_(M) ofsaid structural structural stress σ_(s) in said fatigue-prone region byperforming an operation having a result substantially equivalent to aresult of the following first integration$\sigma_{M} = {\frac{1}{t}{\int_{0}^{t}{{\sigma_{X}(y)}\quad{\mathbb{d}y}}}}$where σ_(x)(y) represents said through-thickness stress distribution andt corresponds to the thickness of said structure; determining a secondcomponent σ_(B) of said structural stress σ_(s) in said fatigue-proneregion by performing an operation having a result substantiallyequivalent to a solution of the following equation for σ_(B)${{\left( \frac{t^{2}}{2} \right)\sigma_{M}} + {\left( \frac{t^{2}}{6} \right)\sigma_{B}}} = {{\int_{0}^{t}{{\sigma_{X}(y)}y\quad{\mathbb{d}y}}} + {\delta{\int_{0}^{t}{{\tau_{xy}(y)}\quad{\mathbb{d}y}}}}}$where y corresponds to a position along said selected cross section, tcorresponds to a thickness of said structure, 6 is a value defined insaid representation of said structure, σ_(x)(y) represents saidthrough-thickness stress distribution, and τ_(xy)(y) represents athrough-thickness shear stress distribution of said structure; andcalculating said structural stress σ_(s) by combining said firstcomponent σ_(M) of said structural stress and said second componentσ_(B) of said structural stress.
 7. A computer-readable medium encodedwith a computer program for analyzing structural stress σ_(s) in afatigue-prone region of a structure according to the method of claim 6.8. A system for analyzing structural stress σ_(s) in a fatigue-proneregion of a structure according to the method of claim 6.